We introduce the new concept of the chromatic sum of a graph G, the smallest possible total among all proper colorings of G using natural numbers. We show that computing the chromatic sum for arbitrary graphs is an NP-complete problem. Indeed, a polynomial algorithm for the chromatic sum would be easily modified to compute the chromatic number. Even forâ€¦ (More)

general, any n-sequence determines a multiset of (= n!/k!(n-k)!) k-sequences. "J Suppose, conversely, that we are given a multiset 01 k-sequences. Does that multiset come from some sequence? If so, is the source sequence unique? Such questions are reminiscent of the problem of reconstructing graphs from vertex-deleted sub raphs, which has received a greatâ€¦ (More)

Dissimilar vertices whose removal leaves isomorphic subgraphs are called pseudosimilar. We construct infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex. Potential impact on the Reconstruction Conjecture is considered. We also construct, for each n, graphs containing a subset of n verticesâ€¦ (More)

The generalized Petersen graph P(n, k) has vertex set V= {u,,, ul, . . . . u,,, 00, Vl, a.., v,-r} and edge set E= {u,u,+r, u,v,, v,v,+~~ for O<i<n1 with indices taken modulo n}. The classification of the Hamiltonicity of generalized Petersen graphs was begun by Watkins, continued by Bondy and Bannai, and completed by Alspach. We now determine the preciseâ€¦ (More)